A log-Sobolev inequality for the multislice, with applications
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Let κ∈N_+^ℓ satisfy κ_1 + ... + κ_ℓ = n and let U_κ denote the "multislice" of all strings u in [ℓ]^n having exactly κ_i coordinates equal to i, for all i ∈ [ℓ]. Consider the Markov chain on U_κ, where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant ρ_κ for the chain satisfies (ρ_κ)^-1≤ n ∑_i=1^ℓ12_2(4n/κ_i), which is sharp up to constants whenever ℓ is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal--Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan--Szegedy Theorem.
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